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Invited Paper
Non-contact optical control of multiple particles and defects
using holographic optical trapping with phase-only
liquid crystal spatial light modulator
Suman Ananda, b, Rahul P. Trivedia, Gabriel Stockdalea, Ivan I. Smalyukha,*
Department of Physics, University of Colorado at Boulder, Colorado 80309-0390, USA
b
Amity Institute of Nanotechnology, Amity University, Noida, Uttar Pradesh 201301, India
a
ABSTRACT
In this work, three-dimensional manipulation of multiple defects and structures is performed in the framework of
holographic optical trapping approach using a spatial light modulator. A holographic optical tweezers system is
constructed using a liquid crystal spatial light modulator to generate multiple optical traps. We optimize the tweezers
setup to perform polarization-sensitive holographic optical trapping and then explore properties of optical trapping in
thermotropic liquid crystals and compare them to the case of isotropic fluids. One of the major challenges complicating
the quantitative measurements in these fluids is the anisotropic nature of the liquid crystal medium, which makes the
tight focusing of the laser beam difficult and considerably weakens optical trapping forces. Using liquid crystals with
low birefringence allows us to mitigate these artefacts. Optical trapping forces and the trap stiffness are first calibrated
for different laser powers using viscous drag forces. This is then used to probe inter-particle and defect-particle
interaction forces as well as to characterize tension of line defects in the bulk of liquid crystals.
Keywords: Optical tweezers, micro-manipulation, multiple traps, liquid crystals, spatial light modulator (SLM), CGH
(computer generated holograms), holography, disclination
1. INTRODUCTION
Non-contact optical control is of great interest for many areas of science and technology as it can allow one to
manipulate objects spanning from single atoms to microscopic particles. Soon after the demonstration of a single beam
optical technique for trapping micrometer-sized transparent dielectric particles suspended in a fluid medium [1], the socalled optical tweezers permeated research in the biological and condensed matter physical sciences. In biology, the
growing research in cellular and molecular biology demanded methods for controlling cells, cellular organelles and
biopolymers, encouraging the optimization and further development of optical tweezers [2, 3]. In physics too, the study
of phenomena at the microscopic scale has become an important area of research. When studied using model systems of
optically-manipulated multiple colloidal particles, phenomena such as self-assembly and many-body interactions provide
an accessible testing ground for ideas about matter at the atomic scale [4], and are interesting in their own right as
complex systems. The need to control and perturb these systems has also driven optical tweezers research [5-8].
Optical tweezers are generated by strongly focusing a laser beam, thus creating an optical gradient that traps transparent
dielectric particles of sizes ranging from micrometric samples down to nanometre scale [1]. Radiation pressure due to
absorption and backscattering competes with the gradient force pulling the object to the focused beam and tends to push
particles downstream. Because of this, stable three dimensional (3D) trapping in a single beam of light is possible only if
the axial intensity gradients are large enough to overcome the radiation pressure. Traditionally, particles which reflect,
absorb, or have an index of refraction lower than that of the surrounding medium have not been trapped. However, the
possibility of generating optical vortices using holographic optical tweezers renders it possible to trap all of these
particles also, a task thitherto unachievable [9].
*
[email protected]; phone: 303-492-7277 (office), 303-492-2898 (Fax), http://spot.colorado.edu/~smalyukh
Emerging Liquid Crystal Technologies IV, edited by Liang-Chy Chien, Ming Hsien Wu,
Proc. of SPIE Vol. 7232, 723208 · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.807873
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Single beam optical tweezers provided scientists with a “single hand” to trap and manipulate an individual particle,
however, the need for “multiple hands” for manipulating many experimental components became clear. The
development of holographic optical tweezers (HOT) was hence a natural step forward along the direction. Within the
last ten years optical tweezers have been revolutionized by the availability of programmable spatial light modulators
(SLM) [10]. In HOT, the SLM is configured to act as a phase-only diffractive optical element allowing a single incident
laser beam to be split into many beams, each forming an individual optical trap. By updating the SLM with a sequence
of holograms, the individual optical traps can be moved in all three dimensions (3D) [10]. Considerable work has been
reported on forming 3D arrays of particles and also modifying the profile of individual traps through appropriate
hologram design [11]. Unlike the other approaches to create multiple tweezers that use scanning mirrors or acousto-optic
deflectors to laterally scan the laser between traps [12, 13], the SLM can manipulate position of objects along the vertical
axis; this is done by generating holograms that act as additional lenses allowing for axial displacement of particle.
Moreover, both lateral and axial displacements of the multiple traps can be set independent of each other to create
complicated 3D structures. Thus, the major advantage of the SLM-based HOT approach over the non-holographic
trapping methods is the possibility of having large number (~200) of traps, each independently controllable not only in
the lateral plane but also along the optical axis of the microscope.
Recently, there has been a growing interest to manipulate colloidal particles immersed in complex fluids such as liquid
crystals (LCs) [14-21], defects in Langmuir monolayers [22], textures [23], and LC droplets [24, 25]. LCs are usually
composed of anisometric molecules, flowing like ordinary liquids, but exhibiting long range orientational order along
with varying degrees of positional order, similar to solid crystals. The average local orientation of the LC molecules is
described by the so-called LC director . From the optical standpoint, an aligned LC is a uniaxial monocrystal with an
optic axis along . Trapping of a colloidal particle in the LC medium is somewhat more complicated than in isotropic
fluids because the refractive index difference between the particle and the host medium depends on the local LC director
[14-21]. Because of the richness of observed phenomena and fascinating experimental capabilities, optical trapping
shows a great promise to become one of the mainstream techniques in the LC studies. Especially when combined with
confocal microscopy imaging in 3D [26-29], the HOT approach opens up new arenas for manipulation and simultaneous
imaging of colloidal particles, defects and structures in liquid crystalline environment [30]. Fluorescence confocal
microscopy has become an established 3D imaging technique used in the biological and medical sciences [25, 26] for
quite some time whose advantages for application in the LC research have been explored recently [28, 29]; however, the
feasibility of its use in combination with laser trapping in anisotropic fluids has not been explored.
The primary focus of this paper is to describe the design, calibration and performance of a HOT system based on a LC
SLM. The HOT setup has been optimized for polarization-sensitive trapping and integrated with a fluorescence confocal
microscope. We describe HOT system designed to automatically manipulate and assemble colloidal micro particles in
water and in LC. We show that the defect lines of different types in LCs can be manipulated using the HOT. The
Fluorescence Confocal Polarizing Microscopy (FCPM) [30] allows us to visualize the 3D director fields as well as the
spatial positions of particles and defects. We employ the 3D manipulation of the optically trapped colloidal particles and
visualize the resultant 3D spatial movement and director structures using FCPM. This setup for simultaneous 3D
manipulation and imaging of colloidal suspensions in LCs further expands the available manipulation capabilities for
control of director fields, particles, and structures in LCs.
2. EXPERIMENTAL SETUP AND MATERIALS
We have constructed a HOT setup (operating at wavelength 1064nm) and integrated it with a fluorescence confocal
polarising microscope (fluorescence confocal microscope FV300 with an inverted base IX 81, Olympus). A detailed
diagram of the optical layout is presented in Fig. 1(a) and the photograph of the assembled system is shown in Fig.1 (b).
All the optical components are mounted on a vibration-isolated optical table (Newport) to provide high degree of
stability to the holographic setup. We use two Olympus oil immersion microscope objectives for the holographic
trapping experiments: 60X (NA ~1.42, ~70% transmission at 1064nm) and 100X (NA ~1.4, ~60% transmission at
1064nm) whose back aperture diameters are 5.62 mm and 3.32 mm, respectively.
We employ a continuous wave, Ytterbium-doped fiber laser (YLM series, IPG Photonics, maximum power 10W) at
1064 nm with the beam diameter of 5.0 mm. The incoming laser beam was made linearly polarized using a Glan-Laser
polarizer (GLP) and its polarization direction was rotated using a half waveplate in order to optimize the efficiency of
the phase modulation. For successful operation of the HOT setup, the beam diameter of the laser has to be first expanded
to overfill the active area of the SLM and then reduced to the size of the back aperture of the objective.
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SLM
Objective
100X
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Fig. 11 (a) Schematic
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(FV300) that
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In order to achieve this, two telescopes are included in the optical train of the HOT setup, one before, and the other after
the SLM. The first 2.5:1 telescope arrangement is used to expand the laser beam from 5.0 mm to 12.0mm in order to
overfill the active area of the SLM. It consists of two identical Plano convex lenses, L1 (100mm) and L2 (250mm),
placed at distance equal to the sum of their focal lengths, so that an expanded collimated beam is produced. By placing
these lenses with their flat surfaces facing one another, spherical aberrations are minimized without resorting to
expensive aplanatic lenses.
We use a reflective, electrically-addressed, phase-only, LC SLM (512 x 512 pixels with a pixel size of 15 x 15 μm2) that
is obtained from Boulder Nonlinear Systems (BNS). This SLM can produce multiple optical traps and control the phase
of the incoming light beam on a pixel-by-pixel basis by modulating it according to the holographic pattern supplied by
the computer. This modulated light beam is imaged at the back aperture of the microscope objective acting as a Fourier
transform lens. Each pixel on the SLM can spatially modulate the local phase of the incident beam in the range of 0 to
2 at the operating wavelength of 1064nm with an effective refresh rate of 10-30 Hz for the entire array. We generated
the holograms making use of a user-friendly graphical software interface HOTkit (obtained from Arryx). Also, the
mirror M1 is used to fold the collimated beam coming from the first beam expander telescope, such that the total angle
between the incident and the reflected beam off the SLM is ~8°. The ideal situation for the SLM to achieve the best
phase modulation characteristics (without any amplitude modulation) is to have normal incidence but this would require
a polarizing beam splitter to be placed in front of the SLM, thus, increasing losses in the system.
A second telescopic imaging system is placed directly after the SLM to reduce the beam size such that it slightly
overfills the back aperture of the microscope objective. This 2.1:1 telescope in the so-called 4f arrangement is
constructed using lenses L3 (850mm) and L4 (400mm) and reduces the diameter of the collimated laser beam (reflected
from the SLM) from 9.0mm to 4.2mm to slightly overfill the back aperture of the microscope objective (back aperture
diameter is 3.32 mm for 100X). The spacing between lenses L4 and L3 is fine-tuned to achieve a collimated output at
the back aperture of the objective. This is done by temporarily removing the objective lens and projecting the laser beam
over a distance of 1-2 m. The distance between the lens L3 and SLM is kept equal to ~85cm (focal length of L3), the
lenses L3 & L4 are kept at 125cm apart (i.e., at distance equal to the sum of focal lengths (f4+ f3) and finally, the
distance between the back aperture of the objective and the position of L4 is equal to ~40cm (the focal length of L4).
The total distance from the back aperture of the objective to the SLM is equal to 2(f4+f3) ~ 250cm. Also, since the
height of the optical train is lower than the height of the entry port of the microscope, it is necessary to increase the beam
height. This is achieved by constructing a periscope using mirrors M3 & M4 (located between the lenses L3 & L4).
In the integrated imaging and manipulation setup, the same objective is used for optical trapping as well as fluorescence
confocal microscopy. In both cases, the laser beams are focussed to a tiny region < 1μm3. A dichroic mirror (Chroma) is
mounted in a rotating filter turret positioned below the microscope objective. The 1064nm IR laser beam entering the
microscope is reflected by this dichroic mirror into the objective lens while the visible light (including the visible laser
beams used for dye excitation in the confocal microscope) is transmitted through it to the charge-coupled device (CCD)
camera and the confocal microscopy scanning head for imaging. The selected camera (Ptgrey, Flea 2, IEEE 1394b,
80frames per second, compatible with the Arryx software) is placed on the vertical arm of the trinocular tube.
Simultaneously with optical trapping, we also determine the LC director configuration in the imaged area, using the
florescence signal from the dye with which the LC is doped. Similarly the positions of the particles are determined using
the fluorescence from a different dye with which they are tagged. Moreover, both probing the locations of particles and
imaging of the director structures in the LC can be done simultaneously by using different imaging channels. The
confocal microscope has the capability to image in three detection channels. It has in total five laser lines for dye
excitation (He-Ne 633nm, He-Ne 543nm, and a 3-line Argon laser operating at 458, 488 and 515nm). An appropriate
combination of laser line and imaging channel can be selected depending on the fluorescence characteristics of the dye.
The dye used to study the samples was excited at 488nm and fluorescence emission was detected in the first channel
(with a low wavelength pass filter at 510nm).
For optical trapping of polystyrene particles (refractive index 1.59) in water, we first prepare a 30 μm thick cell having a
6-7 mm wide channel across the centre of the two confining microscope cover slides. By gluing the glass plates together
with a UV-curable adhesive that has dispersed 30 μm spherical spacers, we assure uniform cell gap thickness of about 30
μm. A sample containing 2μm polystyrene spheres in deionized water is prepared by placing a ~15μL drop of the
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solution at the edge of the cover slide of this cell next to the channel; the cell is then self-filled due to the action of
capillary forces. The typical concentration of polystyrene spheres in the deionized water is kept below 0.01wt%. The LC
cells are assembled from glass plates coated with polyvinyl alcohol (obtained from Alfa Aesar) alignment layers buffed
to set the uniform in-plane far-field director . The thickness of the LC cell is set at 30-60μm using UV-curable adhesive
with the dispersed appropriately sized glass spacers. The samples are prepared using low-birefringence (n~0.04)
nematic LC ZLI-2806 having the average refractive index of ~1.49 (no= 1.48 and ne= 1.52 are ordinary and extraordinary
refractive indices, respectively). To obtain the cholesteric LC, ZLI-2806 is doped with a chiral additive CB15 (obtained
from Merck). For the FCPM studies, the LCs are doped with ~0.01wt% of the fluorescent dye N, N'-Bis(2,5-di-tertbutylphenyl)-3,4,9,10-perylenedicarboximide (BTBP, with fluorescence emission in the range 510-550nm). We use
melamine resin (MR) spherical particles (diameter 3-7 μm) of refractive index ~ 1.68 tagged with the Rhodamine B dye
(with fluorescence emission in the spectral range 585-650nm). The LC mixed with well-separated MR particles is
introduced into the cell by the capillary forces when heated to the isotropic phase of the LC material (to avoid the effects
of flow on the LC alignment). The cholesteric pitch of the used chiral nematic LCs is varied within 5-10 μm by adding
different amounts of the chiral agent (up to 5 wt.%). All used materials are transparent at the trapping wavelength
1064nm, so that no significant laser-induced sample heating (>1°C) is observed. After the material is brought into the
cholesteric or nematic phase by temperature quenching from the isotropic state, the dispersed particles and defect lines
were trapped and studied using the HOT.
3. SETUP ALIGNMENT, OPTIMIZATION AND CALIBRATION
3.1 Alignment
A guide beam at 633nm (part of the IPG Photonics laser source) collinear with the beam at 1064nm is used for
alignment. We use this guide beam for the alignment of the two telescopes which are critical elements determining the
trap quality. Before inserting the lenses in the cage assembly (forming the first telescope) the beam was aligned (using
pinholes) such that it coincides with the principal axis of the optical arrangement along the entire cage assembly. The
lenses L1 and L2 are then mounted in the cage assembly on x-y translation mounts. The distance between these two
lenses affects the trapping system in the following manner. As L1 is pushed towards L2 (from the median position
wherein they are separated by the distance equal to the sum of their focal lengths), the beam exiting the telescope
becomes divergent. This pushes the focal spot away from the objective and deeper into the specimen. Likewise, as L1 is
pulled away from L2, the beam from the telescope becomes convergent, bringing the focus towards the objective.
Movement of lens L1 in the plane perpendicular to the optical axis produces a deflection in the light leaving the
telescope. The second telescope consists of lenses with large focal lengths to reduce aberrations in the system. The
beam is aligned by steering the mirrors M4 and M5 so that it not only remains centered at the back aperture of the
microscope objective but also enters parallel to the microscope optical axis.
To test the quality of alignment in the setup, we use the Airy pattern approach. When a glass slide is placed on the
microscope sample holder, the observation of the back-reflected Airy patterns confirms the diffraction limited nature of
the obtained laser trap. The microscope focus and the beam alignment are tuned until a centrally-placed symmetric Airy
disc with minimized radius is obtained. In the aligned experimental setup (Fig. 1), as the fine focus of the microscope is
varied from 0 to 20μm, the Airy pattern is observed to expand and contract symmetrically (Fig. 2), thus, confirming
good alignment quality.
a
c
b
1 μm
5 μm
e
d
10 μm
15 μm
20 μm
Fig. 2 (a-e) Images show the expanding Airy disc observed upon varying the focus.
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z
3.2 In-plane spatial calibration
For the in-plane spatial calibration, we use the Arryx HOTgui software and calibrate distances as needed for the
holographic optical trapping. There are two calibration procedures that we perform by using 60X microscope objective:
Ruler Calibration and HOT calibration. Ruler calibration allows one to match distances in images obtained using the
camera (in pixels) with the known distances on the ruler slide (in microns). This calibration procedure enables the
software to correctly calculate trap paths and ensures that length measurements given in microns are accurate.
.1
p1 .I'p
0
II
. =0
Fig. 3 HOT calibration via locating the calibration points at known spatial positions.
The HOT calibration enables the software to correctly position optical traps in the microscope field of view. It was done
by using a mirror slide and the SLM-generated focused laser beams at the focal plane of the objective. Figure 3 shows
the diffracted spots matched to the software-generated trap locations, as obtained after the calibration for the HOT setup
(Fig. 1). The calibrated diffraction spots should appear at the largest possible distance from each other while still being
within the field of view. The polarization of the beam before the SLM was adjusted and optimized using a half wave
plate in such a manner that the zeroth order diffraction spot was blocked.
3.3 Power Calibration
Before the trapping experiments, the laser power at the microscope objective and the SLM is measured to help
quantifying the trapping experiments. The measurement of the laser power at the SLM is important to assure that the
power does not exceed the critical laser intensity up to which the SLM operation is not influenced by the high-power
laser beam. The laser power is determined immediately after the SLM by placing a glass cover slide (at 45° to the
reflected beam axis) and measuring the fraction of light reflected off the cover slide using a power meter (1916 C,
Newport). It is ensured that the beam enters the power meter directly and the meter is positioned for maximum intensity
reading; an additional lens is used to contract the beam to the size of the active area of the power meter (vide Fig.1).
Figure 4(a) shows the laser power immediately after the SLM.
To measure the power reaching the microscope objective, the objective is removed and an optical power meter is placed
on the microscope stage and aligned to obtain the maximum laser power readings. The calibration graph for the used
HOT setup is shown in Fig 4(b). As discussed before, due to the relatively poor transmission of the objectives (~70% for
60X and ~60% for 100X) at the operating wavelength, the laser intensity at the sample will depend on the objective and
can be estimated using calibration graphs similar to that in Fig. 4b and known transmission coefficient of the objective
used.
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500 450 400 350 300 250 200 -
a
b
15010000
0.5
1.5
1
Laser power W)
Fig.4 Laser power calibration plots: power at (a) the SLM and (b) at the microscope objective
vs. the output power of the used 1064nm IR laser source.
3.4 Trapping depth calibration
A cell filled with deionized water containing 2μm polystyrene spheres is used for the trapping depth calibration. A single
trap is created using the software. When a polystyrene sphere is brought close to the trap, it experiences the influence of
optical forces, eventually hops towards the trap and becomes spatially-localized, i.e., its Brownian motion is ceased. If
the sphere is trapped in the beam but appears out of focus, it indicates that the trapping plane and the imaging plane are
not coincident. In this case, the lenses L1 and L2 of the setup in Fig. 1 are adjusted to make the imaging and trapping
focal planes coincident.
a
c
b
1 μm
10 μm
5 μm
e
d
15 μm
20 μm
Fig 5. (a-e) Trapping depth calibration using 2μm polystyrene spheres trapped in water (left and right) and a sphere stuck to the
cell surface (centre). When the microscope objective is moved to five different depths (1, 5, 10, 15 and 20μm) along the
microscope’s optical axis, the trapped spheres always remain in focus while the stuck sphere becomes out of focus.
After this is achieved, the SLM should then allow one to adjust the depth of the trapping site to be above, below, or
within the imaging plane. If the SLM-assisted refocusing is not efficient and the spheres can be trapped only in the
imaging plane of the objective but not above and below this plane, this points into the possibility of under-filling of the
microscope objective by the incoming laser beam. In this case, the positions of the lenses L3 & L4 or L1 & L2 are
sequentially adjusted until the 3D trapping is enabled. Figure 5 shows how two freely suspended particles and a particle
stuck to the surface are used for the purpose of calibration of the trapping depth and distances along the optical axis of
the microscope. The functionality of the HOT system to trap in 3D can be verified by trapping a sphere and moving the
objective of the microscope upward or downward. Figure 5 shows images of polystyrene spheres being manipulated
along the vertical direction. The central sphere is stuck to the slide and the particles to its left and right side are trapped.
When the microscope objective is moved from the initial position at 0μm to the new one at 25μm along the vertical
direction, the free-floating colloidal particles remain trapped and in focus whereas the surface-stuck particle becomes out
of focus.
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After the verification of the axial alignment, the setup is further tested for robustness of trapping at different trapping
depths and feasibility of programmed movement of particles in both positive and negative vertical z-direction. In Fig. 6,
the particle in the center is stuck to the cover slip and the spheres at the left and right are trapped and moved above and
below the focal plane, respectively. As can be seen, the appearance of the particle stuck to the cover slide does not
change (always slightly out of focus) whereas the appearance of the two trapped particles changes as they are moved to
either side of the image plane. This robust trapping at different depths along the optical axis of the microscope confirms
that the axial alignment is good and that the image plane and trapping plane are nearly coincident. Moving the objective
upward and downward using the stepper motor with pre-calibrated positioning in the vertical direction, we match axial
displacements of surface-stuck and free-floating particles with those obtained using the software-controlled SLM. This
calibration procedure allows one to assure that distance calibration in the direction along the microscope’s axis is also
appropriate.
a
c
b
0μm
0μm
+5μm
-5μm
e
d
+10μm
-10μm
f
a
+15μm
-15μm
+20μm
-20μm
+25μm
-25μm
Fig 6. (a-f) Trapping of 2μm polystyrene spheres and their motion in positive and negative directions
perpendicular to the image plane. The particle at the centre is stuck to the cover slip and remains at the same
focus. The distances between the trapped particles and the focal plane are marked on the images (a-f).
3.5 Trapping Force Calibration
Characterization of trapping forces is extremely useful in quantitative studies of colloidal interactions, line tension of
defects in LCs etc. [16, 19, 20, 31]. We have performed calibration of optical trapping forces by using the known
viscous drag forces exerted by the LC [19]. By dragging a trapped particle at different trap motion velocities, we find the
critical velocity at which the particle escapes the trap (the so-called escape velocity). At this velocity, the viscous drag
force is equal to the maximum value of the optical trapping force for the given laser power. For this, we made the
trapped particles move repeatedly along the same linear path, each time with an increasingly higher velocity (equal to the
respective velocity of HOT-generated laser trap), as programmed using the software.
We use the Stokes’ formula to calculate the viscous drag force,
, where
is the effective
diameter of the LC colloid (which is larger than the solid sphere with tangential surface anchoring because the bead is
surrounded by director distortions and two surface point defects, the boojums, at the two opposite poles along the farfield LC director),
is the effective viscosity coefficient, and
is the experimentally-determined escape velocity.
Figure 7 shows that the escape velocity
, the respective drag force, and thus also the laser trapping force are
monotonically increasing functions of applied laser power. Note that the slight departure from the linear dependence (for
the direction along rubbing, Fig.7) on laser power might be related to the laser-induced realignment at high laser power
values. The measured viscous drag forces corresponding to the escape velocity give the rough estimate of the respective
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esrape velocity
- 30
t drag force
Parallel to the
rubbing direction
20
15
a
10
4
Perpendicular to the
rubbing direction
0
0
100
120
140
160
220
200
1110
240
260
300
1110
Power at the objective 1mw)
Fig. 7 Escape velocity and the respective drag forces as function of the laser power. We see
directional anisotropy in both measured quantities.
, i.e.,
maximum laser trapping force for a given laser power (for such estimates, one can assume that
equal to the average viscosity of ZLI 2806 at 20°C). The escape velocity measurements were performed along two
mutually orthogonal paths, one parallel and the other perpendicular to the rubbing direction of the alignment layer. We
observe directional anisotropy of the viscous drag forces, Fig. 7, which is related to the anisotropy of
and
.
4. RESULTS AND DISCUSSION
4.1 Spatial rearrangement of multiple particles
As already described, the holographic tweezers are capable of generating multiple traps which can be moved
independent of each other. Multiple movement paths can be generated for each trap in the lateral plane. Figure 8 shows
an example of trapping six polystyrene particles (2 μm) in a hexagonal arrangement and their programmed movement in
the lateral plane. Figure 9 shows particles trapped and programmed to adopt different two-dimensional structures.
a
c
b
o
o
0
0
0
0
0
0
o
o
0
0
0
0
0
0
Fig 8 (a-c) Trapping and movement of six polystyrene particles in water.
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a
c
b
c
OCo
000
o
0
g
f
e
d
o
0
0
0
0
0
0
0
0
0
0
C
0
Fig. 9(a-g) A sequence of images showing simultaneous trapping of polystyrene spheres (2μm in diameter)
in water when organized into structures changing from (a-c) circle to square and when (d-g) particles in the
vertices of two hexagons are programmed to move in different directions.
0
a
0
0
00 000
0
b
c
Fig. 10 (a-c) Trapping and movement of three melamine resin particles in a liquid crystal using HOT.
In general, the optical trapping forces in the LCs (e.g. ZLI-2806) are somewhat weaker as compared to the forces acting
on the particles of the same size dispersed in water (partially due to the fact that the difference between the refractive
index of the particle and the surrounding medium is larger in the case of water) but still sufficiently strong to enable
robust laser manipulation. Fig. 10 illustrates the holographic optical trapping and manipulation of 4μm MR particles in
the bulk of the cholesteric LC of pitch about 5 microns using the SLM-based HOT setup.
4.2. Three-dimensional trapping of multiple colloids
Figure 11 shows the movement of multiple 2μm polystyrene spheres in water along the optical axis. The sequence of
images shown therein demonstrates that the spheres can be programmed to move in different groups while being trapped
at different depths in the vertical direction, along the microscope’s optical axis. All the six spheres trapped in the outer
ring of Fig 11(a) are collectively moved to the vertical position z = -20μm, at the same time as the spheres trapped in the
inner ring are collectively moved to z = +10μm and the central sphere is trapped within the objective’s focal plane at z =
0μm. Similarly, as represented in Fig. 11(b), (c) and (d) all the 2μm trapped spheres are moved to different axial
positions (marked on the respective images) simultaneously using our HOT setup and software.
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a
b
+10μm
+10μ
00
-20μ
a
-10μ
-
c
d
+10μ
-
+10μ
+20μ
0μm
Fig. 11(a-d) Three dimensional (3D) trapping of 2 μm polystyrene particles in water along the axial direction. (a)
The six particles in the vertices of the outer hexagon are trapped and moved to z = -20μm, six particle in the inner
hexagon are trapped at z = +10 μm and the centre particle is trapped at z = 0 μm. The outer ring of particles is
moved to (b) z = -10μm, (c) z = +0 μm and z = +20 μm.
-3 μm
+5 μm
+8 μm
-5 μm
Fig. 12 Three-dimensional optical trapping of four melamine resin particles (7 μm in diameter) at different depths in
the bulk of a nematic LC with planar alignment (as marked near the particles).
a
7 μm
b
7 μm
c
7 μm
a.
Fig.13. In situ three dimensional FCPM of 7μm melamine resin particle while optically trapped in cholesteric
Liquid crystal. They are moved in z direction at different heights by using holographic optical tweezers.
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Figure 12 shows the MR particles dispersed in cholesteric LC as they are trapped and optically moved to be localized at
different positions along the vertical direction parallel to the microscope’s axis. The particles produce elastic distortions
in the LC which, along with the particle positions in the cell, are imaged using the FCPM technique and the fluorescence
signal from the BTBP dye. FCPM image in Fig. 13 shows spatial displacements of the trapped particles obtained using
the HOT setup. These displacements in the vertical cross-section (the XZ-plane) and also departures of layers from flat
alignment parallel to the substrates are clearly demonstrated using the FCPM, Fig. 13.
4.3 Optical trapping using Laguerre-Gaussian beams
XZ
As mentioned in the introduction, a further important advantage of holographic optical tweezers based on the LC SLM is
XZ
that one can not only dynamically control
the position of the optical traps in 3D but also vary the type and shape of the
laser trap. For example, the SLM can produce doughnut-shaped light intensity distributions (Laguerre-Gaussian beam)
that are able to trap particles of refractive index that is higher or lower than that of the surrounding medium. Moreover,
compared to the conventional laser traps with Gaussian beams, the traps formed by doughnut-shaped beams have more
rays hitting a particle under a large incidence angles with respect to the microscope axis and thus are showing more
efficient trapping. From the point of view of biological applications, this is important since the sample can be damaged
by high laser power densities. Likewise, in the LC research, high trapping efficiencies are needed to avoid unwanted
laser-induced realignment when doing quantitative study of anisotropic colloidal interaction forces and tension of
ddisclination. Figure 14 shows polystyrene spheres and MR particles trapped in water and LC, respectively. Clearly, the
particles localize in the high-intensity rings generated by the Laguerre-Gaussian vortex beams of different topological
charges (l) using HOT setup shown in Fig. 1. The laser intensity profile in the lateral section of the used vortex beam
(for l =5 and 10) created by the software is shown in Fig 14, (a,d); the diameter of the ring increases with the increasing
the vortex beam’s topological charge (l).
a
a
c
b
w
l=5
f
e
d
l=10
LC
w
LC
Fig 14 (a-f) Trapping of particles by vortex beam. (a, d) images of an vortex beam of topological charges, l= 5 and 10
generated by using HOT, 2μm polystyrene particles trapped in water (w) by l= 5 (b) and l= 10 (e) vortex beam, 4μm
MR trapped in liquid crystal (LC) using vortex beam with l= 5 (c) and l= 10 (f).
4.4 Manipulation of LC defects
LC
W
LC
The study of topological defects is important for understanding defect-mediated phase transitions in condensed matter; it
is also of a great current interest in cosmology. The nematic LCs are convenient model systems for the study of defects
and can be useful for the “cosmology in laboratory” experiments [30, 34-38]. Long-range orientational order in nematic
LCs can be locally broken (so that the director field
can not be defined) on a point, along a line, or along a wall,
giving rise to defects. The line defects in nematics are called disclinations; they are classified according to their strength
that the director
makes around the defect core when one
m that is defined as a number of revolutions by
circumnavigates the core once. While the half-integer defect lines are topologically stable, the integer-strength
disclinations are not and usually relax into nonsingular configurations of
. In chiral LCs, half-integer defect lines
often have non-singular defect cores (the so called -disclinations LC molecules in the defect core aligned parallel to the
defect line) and form clusters that can be topologically unstable as well. LC defects can be obtained by a rapid quench
from high-symmetry isotropic to the lower-symmetry nematic phase. However, it is usually rather difficult to spatially
control (manipulate) them, a capability that would be desired for the topological study and “cosmology-in-laboratory”
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experiments. Although the field-induced propagation [32] and deformation [33] dynamics of disclinations allow one for
a limited spatial manipulation of these defects, this can be done only at certain boundary conditions. Robust approaches
to manipulate defects are in a great demand and the conventional laser tweezers with polarization control have already
been successfully used for this purpose [30].
In this work, we demonstrate that the defect lines can be also manipulated by using handles in the form of particle
optically trapped and manipulated with the HOT setup. Fig. 15 (a-e) shows manipulation of a topologically stable
disclination using a trapped MR particle with tangential surface anchoring. The particle is slowly pulled in the direction
perpendicular the defect line using a laser trap generated with the HOT. Since the particle motion across the defect is
resisted because of incompatibility of the surface boundary conditions at the particle’s surface and the molecular
alignment within the defect core, the defect line can be stretched similar to an elastic string. Figure 15 (a-d) demonstrates
that the defect line is moved and bent around the particle using optical tweezers. Using a trap and optical trapping force
of the order of 10 pN, we have moved the particle for a distance of ~100 microns along the defect line as shown in Fig.
15 (a-e). After such manipulation, the particle is strongly pinned to the defect line and trapping force of the order of 5070pN had to be used in order to detach the particle from the defect line while leaving this defect line essentially intact.
The defect line was found to straighten (in order to minimize its total energy) and return to its original position after the
particle was separated from it. By stretching the defect line with a colloidal particle and balancing the laser trapping
force with the defect line tension [16] one can also measure tension of defect lines. Similarly to the case of such
experiments performed with time-shared laser traps, calibrated HOT setup also gives values of defect line tension in the
range of 10-100pN, as expected.
a
c
b
d
e
0
Fig 15 (a-d) Manipulation of a disclination by the optically trapped MR particle (4μm in diameter)
(e) movement of particle along the defect line.
One of the most commonly observed textures of the cholesteric LCs is the texture containing oily streaks (vide Fig. 16).
HOT setup allows us to manipulate these defects too. A MR particle of 4 microns in diameter is first trapped and then
moved into the core of the oily streak, Fig. 16. Because of periodic rotation of LC molecules along the in-plane helical
axis within the oily streak’s core, the particle cannot easily penetrate through the structure of the defect core in the oily
streaks. Moving the particle perpendicular to the oily streak allows one to move and stretch the entire oily streak (Fig.
15). Clearly, because of the higher defect line tension ~100pN for this particular oily streak), stretching of these defects
is rather different from that of a disclination shown in Fig. 15 and requires higher trapping forces (and use of higher laser
powers).
a
b
c
Fig. 16 (a-c) A MR particle trapped and manipulated with a oily streak in a cholesteric LC.
Figure 17 shows manipulation of a topologically unstable (with total topological charge equal to 0) defect line using a
trapped particle. The sequence of images In Fig. 17 shows that after a certain manipulation using a HOT trap and the
colloidal particle, the defect cluster of non-singular -disclinations of total topological charge equal to 0 can be disrupted
and remain connected to the particle only on one side (Fig.17, e). The part of the defect cluster on the other side of the
particles is disconnected from the bead (Fig. 17c, d), shrinks to minimize the elastic energy and eventually vanishes
(owing to its instability); the particle remains attached to the end of the remaining branch of the defect cluster (vide Fig.
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17e). Thus, manipulation of defect lines with trapped colloidal particles allows one to test topological stability of LC
defect lines.
a
b
c
d
e
Fig 17. Probing topologically unstable quadruple of the so called disclination
5. CONCLUSIONS
We have constructed the holographic laser trapping system integrated with a fluorescence confocal microscope that is
optimized for polarization-sensitive 3D trapping and imaging of director structures, defects, and colloidal particles in the
LC media and other anisotropic fluids. We have provided examples of the simultaneous multiple-particle manipulation
at different depths of the LC and isotropic fluid samples. We also show feasibility of 3D manipulation of defects and
testing of their topological stability using the holographic optical trapping approach. The combined 3D imaging and
trapping in LCs and LC colloidal suspensions further expands the manipulation and imaging capabilities available for
the study of these intriguing anisotropic media.
ACKNOWLEDGEMENT
We acknowledge the support of the Institute for Complex Adaptive Matter (ICAM), International Institute for Complex
Adaptive Matter (I2CAM) and the NSF grants no. DMR 0645461, DMR-0820579 and DMR-0847782 & the University
of Colorado Innovation Seed Grant award. We also wish to express our thanks for discussions held with Dennis Gardner,
Dr. Clayton Lapointe, Dr. Taewoo Lee, Dr. Pratibha Ramarao, and Dr. Andrij Trokhymchuk. We are also thankful to the
representatives of BNS (Anna Linnenberger) and Arryx (Pamela Korda) for technical assistance.
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